2-forms on $S^2$

Question

I've read that the group $H^2_{dR}(S^2)=\mathbb{R}$. If I'm not wrong, this implies that one can build closed 2-forms that are not exact. Can somebody show me an example, please? Thanks!

Answer 1

$$\omega=x\, dy\wedge dz + y\, dz\wedge dx + z\, dx\wedge dy$$

(Baby Rudin, Ch. 10, ex 22)

$$\omega = \sin\theta\,d\theta\wedge d\phi$$ (Exterior derivative of a complicated differential form)

Or any volume form.